Question

Three points $(2,-1,3)$, $(3,-5,1)$ and $(-1,11,9)$ are

• A. Non-collinear
• B. Non-coplanar
• C. Collinear
• D. None of these

Answer 1

Answer: (C)

Let $A$, $B$ and $C$ be three points whose coordinates are $(2,-1,3)$, $(3,-5,1)$ and $(-1,11,9)$ respectively, then
$\overrightarrow{OA}=2\hat{i}-\hat{j}+3\hat{k}$,
$\overrightarrow{OB}=3\hat{i}-5\hat{j}+\hat{k}$
and $\overrightarrow{OC}=-\hat{i}-11\hat{j}+9\hat{k}$
$\therefore \overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}=\left(3\hat{i}-5\hat{j}+\hat{k}\right)-\left(2\hat{i}-\hat{j}+3\hat{k}\right)$
$=\hat{i}-4\hat{j}-2\hat{k}$
$\overrightarrow{AC}=\overrightarrow{OC}-\overrightarrow{OA}=\left(-\hat{i}-11\hat{j}+9\hat{k}\right)-\left(2\hat{i}-\hat{j}+3\hat{k}\right)$
$=-3\hat{i}+12\hat{j}+6\hat{k}$
$\Rightarrow \overrightarrow{AC}=-3\overrightarrow{AB}$
Thus, the vector $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are parallel having the same initial point $A$.
Hence, the points $A$, $B$, $C$ are collinear.